THE s-process IN ROTATING ASYMPTOTIC GIANT BRANCH STARS Falk Herwig. Norbert Langer. and Maria Lugaro

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1 The Astrophysical Journal, 593: , 23 August 2 # 23. The American Astronomical Society. All rights reserved. Printed in U.S.A. THE s-process IN ROTATING ASYMPTOTIC GIANT BRANCH STARS Falk Herwig Department of Physics and Astronomy, University of Victoria, 38 Finnerty Road, Victoria, BC V8P 1A1, Canada; fherwig@uvastro.phys.uvic.ca Norbert Langer Astronomical Institute, Universiteit Utrecht, P.O. Box 8, NL-358 TA Utrecht, Netherlands; n.langer@astro.uu.nl and Maria Lugaro Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 HA, UK; mal@ast.cam.ac.uk Received 22 November 26; accepted 23 May 6 ABSTRACT We model the nucleosynthesis during the thermal pulse phase of a rotating, solar metallicity, asymptotic giant branch (AGB) star of 3 M, which was evolved from a main-sequence model rotating with 25 km s 1 at the stellar equator. Rotationally induced mixing during the thermal pulses produces a layer (2 1 5 M ) on top of the CO core where large amounts of protons and 12 C coexist. With a postprocessing nucleosynthesis and mixing code, we follow the abundance evolution in this layer, in particular that of the neutron source 13 C and of the neutron poison 14 N. In our AGB model mixing persists during the entire interpulse phase because of the steep angular velocity gradient at the core-envelope interface, thereby spreading 14 N over the entire 13 C-rich part of the layer. We follow the neutron production during the interpulse phase and find a resulting maximum neutron exposure of max ¼ :4 mbarn 1, which is too small to produce any significant s-process. In parametric models, we then investigate the combined effects of diffusive overshooting from the convective envelope and rotationally induced mixing. Just adding the overshooting and leaving the rotational mixing unchanged results in a small maximum neutron exposure (.3 mbarn 1 ). Models with overshoot and weaker interpulse mixing as perhaps expected from more slowly rotating stars yield larger neutron exposures. In a model with overshooting without any interpulse mixing a neutron exposure of up to.72 mbarn 1 is obtained, which is larger than required by observations. We conclude that the incorporation of rotationally induced mixing processes has important consequences for the production of heavy elements in AGB stars. While through a distribution of initial rotation rates, it may lead to a natural spread in the neutron exposures obtained in AGB stars of a given mass in general as appears to be required by observations it may moderate the large neutron exposures found in models with diffusive overshoot in particular. Our results suggest that both processes, diffusive overshoot and rotational mixing, may be required to obtain a consistent description of the s-process in AGB stars that fulfills all observational constraints. Finally, we find that mixing due to rotation within our current framework does increase the production of 15 N in the partial mixing zone. However, this increase is not large enough to boost the production of fluorine to the level required by observations. Subject headings: nuclear reactions, nucleosynthesis, abundances stars: AGB and post-agb stars: evolution stars: interiors stars: rotation 1. INTRODUCTION Trans-iron elements are mainly made by neutron capture reactions on 56 Fe seed nuclei. Two processes have been distinguished according to the neutron density at the production site. In the case of the r-process, the n-densities are high (N n > 1 2 cm 3 ), and the timescale of successive n-capture reactions on heavy isotopes is faster than the -decay timescale. Such a sudden high-density neutron burst creates isotopes far away from the valley of -stability in the chart of nuclides, which successively decay back to the stable isotopes. In contrast, the s-process is characterized by lower neutron densities (N n d1 1 cm 3 ). Neutron captures are generally followed by -decays since unstable isotopes on the s-process path have typical lifetimes of the order of hours. In some cases, however, the unstable isotopes involved have longer lifetimes, and, depending on the neutron density and temperature conditions, branchings can 156 be open on the s-process path, leading to the production of neutron-rich isotopes (see Clayton 1968 for an introduction to the s-process). In asymptotic giant branch (AGB) stars, recurrent He-shell flashes (thermal pulses [TPs]) drive a convective zone that temporarily covers the whole region between the H-burning and the He-burning shells (intershell). Here partial He burning produces a high mass fraction (>.25) of 12 C, and the chain 14 N(, ) 18 F( +, ) 18 O(, ) 22 Ne starting on the abundant 14 N from the H-burning ashes produces a relatively large amount of 22 Ne (mass fraction.2). The 22 Ne(, n) 25 Mg reaction as a neutron source for the s-process was suggested by Cameron (196). Temperatures above K are required for that reaction to release a significant amount of neutrons. Stellar models showed that such high temperatures are achieved in intermediate-mass (M ZAMS =M > 4) AGB stars, which were hence proposed as the main site for the production of s-process elements

2 s-process IN ROTATING AGB STARS 157 belonging to the solar main component, i.e., 9 < A < 24 (Iben 1975; Truran & Iben 1977). However, the neutron density produced by 22 Ne burning in TPs is rather high (above 1 11 cm 3 for T ¼ 3:5 1 8 K). This leads to excesses in the neutron-rich nuclides produced by branchings, for example, 86 Kr, 87 Rb, and 96 Zr (see, e.g., Despain 198), in contrast with the great majority of the observations of s-process enhanced stars such as MS, S, and C stars. In S and C stars the Rb/Sr ratio is typically much lower than solar (Lambert et al. 1995; Abia et al. 21), indicating a low neutron density at the s-process site. Also, the Rb abundances observed in 1 AGB members of the massive Galactic globular cluster! Centauri indicate a low neutron density for the s-process (Smith et al. 2). Lambert et al. (1995) reported the zirconium isotopic abundance obtained by spectroscopic observations of the ZrO band heads in M, MS, and S stars and found no evidence of an excess of the neutron-rich 96 Zr, which can be produced in great amount by the s-process when the neutron density exceeds n cm 3. Another problem is that the activation of the 22 Ne(, n) 25 Mg is expected to produce an excess of 25 Mg in stars enriched in s-process elements. Instead, these stars typically have magnesium isotopic abundances in solar proportion (see, e.g., Smith & Lambert 1986; McWilliam & Lambert 1988). Also, other types of observations tend to exclude AGB stars of intermediate mass as the main s-process site. Observations mainly show that MS, S, and C stars have low luminosity (Frogel, Mould, & Blanco 199) and hence low mass. Feast (1989) performed a study of the kinematics of peculiar red giants, including S, SC, and C stars. On the basis of 183 S-SC stars and 463 C stars, he estimated their mean masses to be 1.3 and 1.6 M, respectively, although this estimate needs to be improved. In summary, the observational evidence and the current state of AGB evolution models suggest that the major nuclear production site of the s-process is low-mass AGB stars. In low-mass AGB stars the temperature in the intershell is not high enough to burn a significant amount of 22 Ne. The 13 C(, n) 16 O neutron source reaction, which was suggested by Greenstein (1954) and Cameron (1955) and is activated at lower temperatures (:9 1 8 K), is expected to play the major role (Iben & Renzini 1982; Gallino et al. 1988; Käppeler et al. 199). However, an amount of 13 C higher than that present in the H-burning ashes is needed to reproduce the observed enhancements of heavy elements. In order to form a sufficient amount of 13 C, it is hence speculated that some protons mix into the 12 C-rich intershell (see Busso, Gallino, & Wasserburg 1999 for a general review of the s-process in AGB stars). In recent years a picture of the s-process based on these results has emerged (Gallino et al. 1998), and it is summarized in Figure 1. The He flash convection zone homogenizes the intershell region, and 12 C produced in the He-burning shell is mixed up to just below the location of the H-burning shell. The dashed line in Figure 1 indicates that, after the convective pulse is extinguished, the convectively unstable envelope may extend down into H-free layers of the intershell region. This phenomenon allows processed intershell material to be carried into the envelope and hence to the stellar surface (third dredge-up). At the end of third dredgeup a layer is created where the H-rich envelope directly neighbors the 12 C-rich intershell. This layer is a favorable region for the formation of a zone where 12 C and protons are partially mixed. As the temperature increases in the region a pocket of 13 C forms by proton captures on 12 C. Subsequently, the 13 C serves as a neutron donor via the reaction 13 C(, n) 16 O, which is activated during the Fig. 1. Schematic representation of the s-process in the interpulse phase of TP-AGB stars in a space-time diagram. The ordinate covers the mass range between the H- and He-burning shells (intershell). The time marks represent a rough estimate.

3 158 HERWIG, LANGER, & LUGARO Vol. 593 following interpulse phase at T K so that neutrons are released under radiative conditions (Straniero et al. 1995). Typically, the s-process occurs on a timescale of several tens of thousands years and before the onset of the following TP. In the convective pulse the 22 Ne neutron source is only marginally activated. It is most reasonable to assume that the H/ 12 C ratio in the partial mixing zone varies somehow continuously from a few hundred in the envelope to zero in the intershell. Then one finds in the top layers of the partial mixing zone another pocket made of 14 N, which forms where the H/ 12 C ratio is larger than where the 13 C pocket forms. Without further mixing the partial mixing layer is strictly stratified during the interpulse period: as shown in Figure 1, the two pockets coexist without much interaction. During the interpulse phase the temperature does not reach values required for the 14 N(, ) 18 F reaction. The 14 N pocket is engulfed into the following He flash convection zone, where it might slightly contribute to the production of 22 Ne. Stellar models that use a standard treatment of convective mixing cannot produce the 13 C pocket because extra mixing processes are required to allow a small amount of protons to enter the 12 C-rich region. Gallino et al. (1998) and following studies by that group and Goriely & Mowlavi (2) have studied the s-process by assuming a certain proton profile extending into the 12 C-rich region without relating explicitly to a specific physical process. In these studies it was implicitly assumed that any subsequent alterations of the abundances in the partial mixing zone are due to nucleosynthesis only and that no mixing takes place during the interpulse phase after the initial formation of the partial mixing zone. In this way it was possible to develop a fairly good understanding of the s-process nucleosynthesis in the 13 C pocket. For example, it was shown that many observations are reproduced with models in which the neutron exposure in the 13 C pocket is up to.4 mbarn 1 at solar metallicity. In order to account for the observed s-process overabundances, the partial mixing zone needs to have a mass of 1 4 to 1 3 M. These s-process models with an assumed H profile for the partial mixing zone can account for many of the overall observed properties of the s-process. However, to explain the observed s-process signature in AGB stars as a function of metallicity, the s-process model described above requires that stars of the same mass and metallicity have different neutron exposures due to the 13 C neutron source during the AGB interpulse phase (Busso et al. 21). A mixing process for the formation and evolution of this neutron source that allows for some spread in the efficiency of producing neutrons seems to be necessary. Van Winckel & Reyniers (2) arrive at the same conclusion from observations of weakly metal-poor, post-agb stars with a 21 lm feature. In these objects the s-process element signatures are easier to measure than in the cool progenitor AGB stars. Also, the spread in the Pb abundance observed in very low-metallicity stars (see, e.g., Van Eck et al. 23) could fit into the current s-process model only if there is a spread in the number of neutrons available in the pocket for the s-process. Such a spread in the efficiency of the 13 C neutron source is also required to explain the measurements of isotopic ratios in single presolar silicon carbide grains (SiC) recovered from pristine carbonaceous meteorites (Zinner 1998). The majority of these grains are believed to have formed in the circumstellar dusty envelopes surrounding carbon stars. The main evidence for this comes from measurements of aggregates of SiC that have shown a very strong signature of the s-process in the heavy elements present in trace (see, e.g., Gallino, Busso, & Lugaro 1997). Nicolussi et al. (1997, 1998a, 1998b) and Savina et al. (23) performed measurements of the composition of Sr, Zr, Mo, and Ba in single SiC grains. Isotopic ratios that are sensitive to the efficiency of the 13 C neutron source have been found to show a large spread within the single-grain data. This can be explained if SiC grains were formed in a multiplicity of low-mass AGB stars with a range of neutron exposures in the 13 C pocket. These isotopic ratios are those involving nuclei with a very low neutron capture cross section, such as nuclei with numbers of neutrons equal or near to nuclear magic numbers, i.e., 88 Sr/ 86 Sr, 9,91,92 Zr/ 94 Zr, and 138 Ba/ 136 Ba. A detailed analysis of SiC grain data indicated that a spread of the order of 5 is necessary in the neutron exposure for a given mass and metallicity (Lugaro et al. 23a). These new insights are specifically important for constraining the physical processes that are responsible for the partial mixing between envelope and core and thereby lead to the formation of the 13 C neutron source. Moreover, neutron capture elements are in general becoming an increasingly important target of stellar observations, for example, of IR observations in planetary nebulae (Dinerstein 21). The open problem addressed in this paper is the role of rotational mixing. Rotation is an effect that has to be taken into account when studying stellar evolution and nucleosynthesis. Most F stars, which are the main-sequence progenitors of low-mass AGB stars, show rotational velocities of a few hundred kilometers per second (Royer et al. 22; Royer, Zorec, & Frémat 23). The importance of rotation as a physical process in AGB stars is not restricted to mixing and nucleosynthesis. Rotation in stars during their late evolutionary phase possibly drives the shaping of bipolar proto planetary nebulae. According to the interacting stellar winds model (Kwok 1982; Balick 1987), a fast ionized stellar wind interacts with an equatorially dense AGB circumstellar envelope that could be the result of inhomogeneities associated with rotation. This model can qualitatively explain the presence of sharp radial structures and the wide variety of shapes found in planetary nebulae (see also Icke, Balick, & Frank 1992). Reimers, Dorfi, & Höfner (2) find that elliptical or weakly bipolar planetary nebula shapes can result from dust-driven winds of rotating AGB stars. Soker (21) considers the possibility that increased rotation in an AGB envelope may result from swallowing another celestial body, like a companion star in a binary system or an orbiting planet. Indeed, this hypothesis gains support from the recent discovery of many extrasolar planets and the detection of water vapor around evolved AGB stars, possibly due to the presence of comets (Melnick et al. 21). García-Segura, Langer, & Różyczka (1999) present hydrodynamical models in which an equatorial density enhancement originates around an intermediate-mass single AGB star with a fast rotating core that can spin up the extended envelope during mixing events associated with the He-shell flashes. Langer et al. (1999) evolved a 3 M stellar model from the main sequence to the AGB phase, including the effects of rotation on the stellar structure and mixing. They found that rotationally induced mixing at the core-envelope interface after a TP could be responsible for the formation of the partial mixing zone that hosts a 13 C pocket and subsequent s-process nucleosynthesis. They also found that mixing in

4 No. 2, 23 s-process IN ROTATING AGB STARS 159 the partial mixing layer continues throughout the entire interpulse phase. In this paper we investigate mixing and the s-process at the core-envelope interface of this stellar model with rotation. We also present a comparison with models including mixing due to hydrodynamic overshooting, as well as parametric models that further illustrate the effect of slow mixing of the s-process layer during the interpulse period. In the current model the s-process occurs in every interpulse-pulse cycle from when third dredge-up starts until the end of the AGB evolution. Several stellar parameters that are important for the computation of the s-process, e.g., the mass of the intershell, the temperature at the base of the convective shell, the overlapping factor between subsequent pulses, and the third dredge-up, are different at each interpulse-pulse cycle. Detailed s-process calculations, such as those of Gallino et al. (1998), take into account these effects. However, the features of the 13 C pocket are kept the same in each interpulse, and the changes in the temperature are not large enough to affect the modality of the burning of 13 Cin different interpulses. The only effect is that in detailed calculations the neutron exposure in the pocket slightly decreases with the interpulse number because the amount of s-process material increases in the intershell. We perform simulations of the s-process over only one interpulse period, which in first approximation represents all the interpulse periods. In x 2 we derive the basic properties of the partial mixing zone and interpulse s-process layer from average observational features and simplified AGB evolution properties. We describe the stellar models and the nucleosynthesis code in x 3. Section 4 is devoted to our scheme for the heavy s-process neutron sink. The properties and effects of mixing induced by hydrodynamic overshoot and by rotation are presented in xx 5 and 6, respectively. Mixing for the s-process is further explored with synthetic models in x 7. The particular problem of the production of 19 F is addressed in x 8, and we present a final discussion in x CONSTRAINTS ON THE PROPERTIES OF THE PARTIAL MIXING ZONE The properties of a partial mixing zone that reproduces the s-process features observed in AGB stars can be studied in detail with models including the effect of many consecutive TPs and neutron exposure events (Gallino et al. 1998; Goriely & Mowlavi 2). Here we derive some basic constraints on the properties of the s-process zone by following a much simpler approach. We consider two s-process indicators: the index s/s is the overproduction factor of s-process elements with respect to the initial solar value. We have used for this index the average of the production factors of Y and Nd. The index ½hs=lsŠ ¼½hs=FeŠ ½ls=FeŠ is the abundance ratio of heavy-to-light s-process elements. We have used ½ls=FeŠ ¼ 1 2 ð½y=fešþ½zr=fešþ and ½hs=FeŠ ¼ 1 5 ð½ba=fešþ½la=fešþ½ce=fešþ½nd=feš þ½sm=fešþ, where square brackets indicate the logarithmic ratio with respect to the solar ratio. Observationally, the spectroscopic studies of the s-process abundances in evolved low-mass stars of solar metallicity can be summarized as < logðs obs =s Þ < 1 and :5 < ½hs=lsŠ < (Busso et al. 1995). The observed overproduction factors in the envelope are related to the overproduction factors in the s-process zone by two dilution factors that result from two subsequent mixing events: the He flash convective mixing and the third dredge-up. Assuming that no significant amount of s-process material is available in the envelope initially and considering only the contribution of the s-process in the interpulse, the abundance of any species in the envelope after third dredge-up events in m identical TP cycles is related to the abundance in the s-process zone (partial mixing zone [PM]) by Y env ¼ qmy PM M PM M DUP M IS M env ; ð1þ where M DUP, M IS, and M env are the masses of the dredgedup layer, the intershell zone covered by the He flash convection, and the envelope, respectively. In low-mass AGB stars with core masses of.6 M these quantities are of the order of M DUP M, M IS 1 2 M,and M env :5 M. These masses vary from pulse to pulse and are dependent on the core mass and the treatment of mixing. For example, in the model sequence with rotation M IS ¼ M at a core mass of M c ¼ :746 M (Langer et al. 1999), while in the sequence with overshooting at all convective boundaries M IS ¼ 2:4 1 2 M at a core mass of M c ¼ :628 M (Herwig 2). The mass M PM refers to the layer of the partial mixing zone that at the end of the interpulse phase contains s-process material. In models without mixing during the interpulse phase this corresponds to the region of the partial mixing zone where initially 3 < log XðHÞ < 2 (Goriely & Mowlavi 2). In models with mixing during the interpulse, as in the case with rotation, the extent of the partial mixing zone can only be determined at the end of the interpulse when the calculation has yielded the s-process nucleosynthesis and mixing result. The factor q describes the effect of overlapping He flash convection zones in subsequent TPs (see below). Without resorting to the detailed results of full stellar evolution calculations, we estimate the number m of TPs with dredge-up events that enrich the envelope in a semiempirical way. As we have discussed in x 1, carbon stars are believed to be the result of recurrent third dredge-up events in lowmass stars with initial zero-age main-sequence (ZAMS) masses predominantly in the range 1:5 < M=M < 3 (see also Groenewegen, van den Hoek, & de Jong 1997). These stars end their lives as white dwarfs with masses in the range :57 < M=M < :68, according to the revised stellar initial-final mass relation of Weidemann (2). In fact, the mass distribution of white dwarfs peaks at or just below.6 M (Koester, Weidemann, & Schulz 1979; Weidemann & Koester 1984; Bergeron, Saffer, & Liebert 1992; Napiwotzki, Green, & Saffer 1999). This mass distribution is very similar to that of central stars of planetary nebulae (Stasińska, Górny, & Tylenda 1997), which are in the evolution phase between AGB stars and white dwarfs. This means that the majority of carbon stars must have achieved the necessary abundance enrichment through the third dredge-up before or when reaching a core mass of.6 M. According to the synthetic AGB models of Marigo, Bressan, & Chiosi (1996), significant dredge-up must commence at core masses of.58 M, or even below in some cases, in order to reproduce the carbon star luminosity function (Marigo, Girardi, & Bressan 1999). Therefore, the relevant chemical enrichment of AGB stars typically occurs within an effective core mass growth of about DM cg ¼ :2 M, and maybe up to.6 M in some cases. For low-mass

5 16 HERWIG, LANGER, & LUGARO Vol. 593 TP-AGB stars, the core mass growth per TP is about DM H M, and with a dredge-up parameter of ¼ :5, about m ¼ 7(DM cg ¼ :2), and possibly up to 2, TPs with dredge-up mixing (if dredge-up starts at M c ¼ :54 M, DM cg ¼ :6) can be considered to be responsible for the abundance enrichment. Equation (1) does not take into account that processed heavy elements accumulate in the intershell from one TP to the next because the He flash convection zone is partly overlapping (overlap factor r) with layers that have been swept by the previous He flash convection zone. In the case of nucleosynthesis of a species (s) during the interpulse phase in a partial mixing zone, the production stays roughly constant from TP to TP. The abundance due to such a production is X s ¼ X PM M PM =M IS. The total intershell abundance at the nth TP with third dredge-up is then given by X n ¼ X s þ rx n 1. The number l of TPs needed to approach a 9% level of some asymptotic value for the intershell abundance of species s is given by l ¼ 1= log r. For n > l, it follows that X l 1 X n X s q ¼ X s r i : The overlap factor decreases as a function of TP number and also depends on the details of the third dredge-up efficiency. Typically, overlap factors in stellar models decrease from about.8 at the earliest TPs to an asymptotic value larger than.4. The condition n > l is approximately satisfied for rd:6, for which equation (2) returns q ¼ 2:3. Other triplets of (r, l, q) are (.4, 3, 1.6) and (.8, 11, 4.6). By evaluating equation (1) for the numbers specified above, and with Y env in equation (1) given by using the maximum logðs obs =s Þ¼1, we derive a logarithmic expression that relates the s-process overabundance in the PM zone with the mass of that zone: i¼ log M PM logðs PM =s Þþc ; where c ¼ :14 (,.44) for m ¼ 1 (7, 2). In Figure 2 we show the variation of logðs PM =s Þ and [hs/ls] PM with the neutron exposure. These trends have been obtained by fully implicit network calculations containing the s-process nucleosynthesis with neutron captures on all isotopes from He to Pb (as in Lugaro et al. 23b) and all relevant charged particle reactions (see x 3 for reaction rate references). As more neutrons are released, the average s-process overabundance increases. In the current s-process model the partial mixing zone by definition does not extend into the He-burning shell, and therefore the partial mixing zone cannot exceed the mass of the intershell layer. Thus, the mass available for the pocket is M PM < 1 2 M. This, together with equation (3) and m ¼ 1, requires that logðs PM =s Þ > 1:86 and translates into a minimum for in the partial mixing zone. In Figure 2 the left shaded part of the diagram with <:2 mbarn 1 is thereby excluded as the predominant region of in the partial mixing zone. Values of logðs PM =s Þ < 1:86 in that region would require a prohibitively large partial mixing zone (M PM > 1 2 M )in order to reproduce the most s-process enriched stars of solar metallicity with logðs obs =s Þ¼1. From ½hs=lsŠ obs <, the -values corresponding to the right shaded area in Figure 2 can be excluded as typical for the s-process. In fact, if the neutron exposure values predominant in the pocket ð2þ ð3þ Fig. 2. The s-process indices logðs=s Þ and [hs/ls] in s-processed material as function of the neutron exposure from a network calculation including all relevant light and heavy elements from H to Pb. The initial 13 C mass fraction is.3, and neutrons are released by the 13 C(, n) reaction at a constant temperature of 9:8 1 7 K. The neutron exposure corresponding to the shaded areas can be excluded as the dominant contribution in the s-process site (see text). As discussed in the Appendix, the presence of light neutron poison like 14 N would limit the neutron exposure that can be achieved, but the relations of and the shown indices are only very weakly affected. exceed.5 mbarn 1, the envelope s-process abundance will show ½hs=lsŠ >. This case is discussed in x 5.3 with an example in Table 1 (see also Lugaro et al. 23b). To summarize, we estimate the following properties for the partial mixing zone in stars of solar metallicity: 1. The 13 C pocket should generate a predominant neutron exposure in the range.2.5 mbarn Using equation (3) and logðs PM =s Þ¼4, corresponding to the upper limit of, the mass of the partial mixing zone should obey M PM > M (assuming m ¼ 1), in agreement with the detailed studies. 3. STELLAR EVOLUTION AND NUCLEOSYNTHESIS MODELS Stellar evolution models with rotation are computed in the same way as in Langer et al. (1999). The onedimensional hydrodynamical stellar evolution code (Langer 1998; Heger, Langer, & Woosley 2) considers angular momentum, the effect of centrifugal force on the stellar structure, and the following rotationally induced transport mechanisms for angular momentum and chemical species: TABLE 1 Effect of the 13 C Pocket Mass on the s-process Efficiency and Abundance Distribution IP BM/1 BM/2 BM a BM 2 [ls/fe] [hs/fe] [hs/ls] a BM: benchmark 13 C pocket corresponding to case with hydrodynamic overshoot with f ¼ :128.

6 No. 2, 23 s-process IN ROTATING AGB STARS 161 Eddington-Sweet circulation, Solberg-Høiland and Goldreich-Schubert-Fricke instability, and dynamical and secular shear instability. The l-gradient, which acts as a barrier for rotationally induced mixing, and the Ledoux criterion for convection and semiconvection are considered. The nuclear energy generation is computed in the operator split approximation with a network including p-p chain, H-burning cycles, and He burning. For comparison, we analyze stellar models with hydrodynamic overshoot of Herwig (2). These models feature a partial mixing zone of protons and 12 C very similar to that assumed for s-process calculations with a parameterized partial mixing zone, as in Goriely & Mowlavi (2). They assume, however, that overshoot is to some extent present at all convective boundaries. The two processes of rotation and overshoot are considered independently from each other since the stellar evolution models with rotation do not include overshoot and the overshoot models do not include rotation. The nuclear reactions under consideration for the s-process do not contribute in a significant way to the energy generation in the star. Hence, postprocessing is a valid approximation and a faster and easier-to-handle alternative to recomputing the whole stellar evolution including all the needed nuclear species. An important feature of our rotating-agb stellar models is a weak and persistent mixing of the partial mixing layer, where the 13 C and 14 N pockets are located next to each other. The s-process cannot be computed under the assumption of stratification anymore, as done, for example, by Lugaro et al. (23b). Therefore, we have developed a nucleosynthesis and mixing postprocessing code (SBM6) that solves simultaneously for mixing processes and charged particle nuclear reactions, as well as neutron production and destruction reactions and -decays. The code uses thermodynamic and mixing properties of the stellar evolution models as input and follows the abundance evolution of the nuclear network described below. In the stellar evolution code with rotation, the time and spatial resolution is determined by the H-burning shell. Grid rezoning after the TP at the core-envelope interface may introduce some numerical diffusion, which can be avoided if the s-process simulation is carried out on a fixed grid beginning right after the formation of the partial mixing zone. In the postprocessing we use a high-resolution, equidistant Lagrangian grid with 4 points to cover our partial mixing layer, which has a mass of 1 4 M. The solution scheme of coupled burning and mixing is fully implicit (Herwig 21) with adaptive time steps. A Newton-Raphson iteration is accepted as a solution if the corresponding greatest relative correction is less than 1 3 for any species at any grid point with a mass fraction X > This guarantees a numerical precision of.1% for any species at each time step and the correct integration of the neutron abundance, which is treated like that of any other species. The neutron density typically encountered during the interpulse (N n < 1 7 n cm 3 ) corresponds to XðnÞ < In practice the precision for neutrons is much better than.1%, even for XðnÞ < 1 21, because the convergence of the set of equations is determined by other species. Because of the fully implicit solution scheme, only a fraction (about 1/1) of the time steps of the evolutionary code need to be computed with the SBM6 code, because in the partial mixing zone the thermodynamic conditions change much more slowly than at the location of the H-burning shell. For the postprocessing simulations, we have considered the following reactions in the SBM6 code: 1. ( p, ) from the NACRE compilation (Angulo et al. 1999) on 12 C, 13 C, 13 N, 14 N, 15 N, 16 O, 17 O, 18 O, 19 F, 2 Ne, 21 Ne, 22 Ne, 22 Na, 23 Na, 24 Mg, 25 Mg (to both 26 Al g and 26 Al m ), 26 Mg, 27 Al, 28 Si, 29 Si, and 3 Si; 31 P and 26 Al g ( p, ) from Iliadis et al. (21); 14 C( p, ) from Wiescher, Goerres, & Thielemann (199); 2. ( p, ) from NACRE on 15 N, 17 O, 18 O, and 19 F; 3. -decays have been assumed to follow instantaneously where applicable, except for the -decays of 22 Na, 26 Al g, 13 N, 14 C, and 59 Ni from the Karlsruher Chart of Nuclides; 4. (, ) from NACRE on 12 C, 15 N, and 18 O, and on 14 C from Gai et al. (1987), Hashimoto et al. (1986), and Funck & Langanke (1989); 1 5. (, n) from NACRE on 13 C and 22 Ne; 6. (n, ) from the compilation by Lugaro (21, Appendix B), which is largely based on Bao et al. (2), on 12 C, 14 N, 16 O, 21 Ne, 2 Ne, 22 Ne, 23 Na, 24 Mg, 25 Mg, 26 Mg, 27 Al, 28 Si, 29 Si, 3 Si, 31 P, 56 Fe, 57 Fe, 58 Fe, 59 Co, 58 Ni, 59 Ni, 6 Ni, 61 Ni, and 62 Ni; 7. (n, p) neutron recycling reactions on 14 N (Gledenov et al. 1995) and 59 Ni from the Reaclib Data Tables of nuclear reaction rates (1991), 2 updated version of the compilation by Thielemann, Arnould, & Truran (1986); 8. Light and a heavy neutron sinks account for neutronabsorbing species between 32 S and 56 Fe and above 62 Ni, respectively (see x 4 for details). We consider the light n-capture reactions that are important to determine the n-density. The efficiency of isotopes to absorb neutrons can be measured by the product of the n-capture, Maxwellian-averaged cross section and the number density of the isotope. Assuming a solar abundance distribution, the 12 most important light n-absorbing isotopes are [in the order of decreasing absorbing efficiency; (n, ) reaction unless noted]: 14 N(n, p), 3 Si, 27 Al, 28 Si, 29 Si, 25 Mg, 16 O, 14 N, 24 Mg, 23 Na, 2 Ne, and 12 C. The order of this list of n-capture isotopes changes at locations in stars where a nonsolar abundance distribution has been established; e.g., in the 14 N pocket (see x 5) the 14 N abundance is so high that the n-captures by 14 N(n, ) reactions outnumber all other (n, ) reactions on light elements and are themselves dwarfed by those of 14 N(n, p) 14 C. The long-lived ground state of the unstable nucleus 26 Al is also a major neutron poison because of its very large cross sections for (n, p) and (n, ) reactions (Skelton, Kavanagh, & Sargood 1987; Koehler et al. 1997). This nucleus is produced by proton captures on 25 Mg during H burning. However, it is efficiently destroyed by neutron captures in the TP convective regions; hence, it is not present in the 13 C pocket, and we have omitted it in the computation. The neutron recycling induced by the 14 N(n, p) 14 C reaction is of particular importance for the s-process in rotating AGB stars. This reaction deprives the Fe seed and transiron elements of available neutrons. The major effect is that the neutron exposure is smaller if 14 N is present (see the Appendix). In addition, the 14 N(n, p) 14 C reaction opens a channel to make the isotope 15 N, and hence potentially 19 F. 14 C has a half-life of 573 yr. Under the typical conditions 1 We have used the electronic NETGEN database to retrieve these and other rates (Jorissen & Goriely 21). 2 See

7 162 HERWIG, LANGER, & LUGARO Vol. 593 of n-production in the interpulse phase (T ¼ K, ¼ 37 g cm 3 ), 14 C can capture an -particle or a proton. The 14 C(, ) 18 O reaction is 2.5 times slower than that of 13 C(, n) 16 O, but still affected by large uncertainties. The p-capture reaction of 14 C[ 14 C( p, ) 15 N] is 3.7 times faster than that of 12 C( p, ) 13 N and about as fast as that of 13 C( p, ) 14 N. No other p- or-capture reactions of 14 C are important, and 14 C(n, ) 15 C can also be neglected. n cap, (L/G) APPROXIMATING THE s-process WITH A NEUTRON SINK We introduce two reactions and two artificial particles that together give an estimate of the overproduction of s-process elements and the s-process distribution. First, we consider the reaction 62 Ni(n, ) 63 G, where 63 G is an artificial particle with mass number 63. We identify the number abundance of 63 G with the combined number abundance of all isotopes heavier than 62 Ni: Yð 63 GÞ¼ X29 A¼63 Yð A SÞ ; where S stands for the element symbol of the respective species with A > 62. For a solar composition, the number abundance of 63 G is Y ð 63 GÞ¼5: Starting on 56 Fe, a suitable neutron exposure will quickly lead to the formation of heavy particles A S (with A > 62); hence, the abundance of 63 G increases. In order to count the number of neutron captures that occur on species with A > 62, we introduce a second reaction and a second artificial particle L, 63 G(n, 1 L) 63 G, which plays the role of heavy neutron sink. The Maxwellian-averaged cross section of 63 G is computed in the usual way (Jorissen & Arnould 1989): ð4þ 2 1 n cap from full s-process network (L/G) from calculation with heavy sink τ / mbarn Fig. 3. Comparison of the s-process parameters n cap and L/G as functions of. The quantity n cap is computed with a full s-process network calculation, while the ratio L/G comes from the calculation in which the s-process is approximated with a sink. The first increase of n cap with respect to L/G at very low is due to neutron captures on isotopes from 56 Fe to 62 Ni. From :5 mbarn 1 the results obtained with the neutron sink show larger and larger differences than the results obtained with the full s-process network. The neutron sink representation is a valid approximation for <:6 mbarn 1, i.e., in the range of interest for solar metallicity stars. the stable neutron magic nuclei on the s-process path ( 88 Sr, 138 Ba, 28 Pb), and the averaged cross section decreases. As more neutrons are released, the contribution of species in between increases and finally dominates the sink cross section. The choice of a constant sink neutron cross section for the entire s-process simulation is the largest individual uncertainty when approximating the s-process with artificial ð 63 GÞ¼Yð 63 X29 1 GÞ A¼63 A Yð A SÞ : ð5þ Neutron captures will occur repeatedly on individual 63 G particles, thereby simulating the production of increasingly heavy s-process isotopes. These neutron captures are responsible for the final s-process element distribution of the species represented by 63 G. They do, however, not change the number abundance of 63 G. The ratio ðl=gþ ¼Yð 1 LÞ=Yð 63 GÞ is similar to the customary quantity n cap (Clayton 1968) and is a measure of the s-process distribution (Fig. 3). In fact, if one defines the artificial sink particle G to be the product of an n-capture on 56 Fe, then ðl=gþ ¼n cap. With our choice of G being the product of a n-capture on 62 Ni, n cap is slightly larger than (L/G) for a given neutron exposure because n cap takes into account the n-captures on isotopes from 56 Fe to 62 Ni. These n-captures mainly take place during the initial phase of the neutron exposure phase for <:1 mbarn 1. The neutron cross section of the sink particle 63 G depends on the abundance distribution of the particles it represents. In Figure 4 the Maxwellian-averaged sink cross section at 8 kev according to equation (5) is shown during calculations of the s-process nucleosynthesis with the network including neutron captures on all isotopes up to Pb, starting with a solar abundance distribution of trans-iron elements. The variation of ( 63 G) reflects the changing abundance distribution of heavy elements. Initially, species accumulate at log N n [cm -3 ] t/yr N n σ( 63 G) Fig. 4. Neutron density (solid lines, left ordinate) and Maxwellianaveraged neutron capture cross section (dashed lines, right ordinate) of sink particle 63 G (described in x 4) as a function of time from s-process calculations as those performed for Fig. 2 with a network including all heavy elements up to Pb. Three test calculations are presented, starting with three different amounts of 13 C mass fractions (indicated by the labels). The variation of the cross section of 63 G reflects its dependence on the abundance distribution of the species represented by the sink particle σ(8kev)/ [mbarn]

8 No. 2, 23 s-process IN ROTATING AGB STARS 163 τ(8kev) full s-process NW σ G =8mbarn σ G =1mbarn σ G =12mbarn sink reactions, as described above. However, it turns out that the error introduced by using a constant sink cross section is sufficiently small for our purpose. In Figure 5 we show the neutron exposure of calculations with the sink treatment with three choices of the sink cross section as compared with s-process calculations performed with a network including neutron captures on all isotopes up to Pb for the three cases of Figure 4. For low neutron exposures, the influence of the sink cross section is small, because the neutron density is dominated by 56 Fe and the lighter neutron capture elements. For the s-process in the partial mixing zone of stars of solar metallicity, the most important range for is between.2 and.5 mbarn 1. We choose for our simulations ð 63 GÞ¼12 mbarn, which reproduces the neutron exposure from the s-process calculation up to Pb within 1%. We also introduced a light neutron sink reaction to take into account neutron captures on elements from S to Mn, which are missing in our network. For this reaction, which plays only a minor role, we have used the value light ð8kevþ¼7:36 mbarn given by Lugaro (21). 5. MIXING FROM HYDRODYNAMIC OVERSHOOT The hydrodynamical properties of convection inevitably result in some turbulent mixing into the stable layers adjacent to convectively unstable regions. In fact, any model of convection in the hydrodynamical framework predicts that the turbulent velocity field decays roughly exponentially inside the stable layers (see, e.g., Xiong 1985; Freytag, Ludwig, & Steffan 1996; Asida & Arnett 2). A depth- and time-dependent hydrodynamic overshoot approximation has been used in a number of recent studies (Herwig et al. 1997; Mowlavi 1999; Mazzitelli, D Antona, & Ventura 1999; Salasnich, Bressan, & Chiosi 1999; Herwig 2; Cristallo et al. 21), with the aim to capture the main t/yr.3.1 Fig. 5. Evolution in time of the neutron exposure for the three calculations shown in Fig. 4, represented by the solid lines (initial 13 C mass fractions indicated by the labels). Each calculation is compared to three tests computed with different values of the Maxwellian-averaged neutron capture cross section of the neutron sink reaction ( 63 G) described in x 4. consequences of hydrodynamic mixing into the stable layers induced by convection Third Dredge-up and H/ 12 C Partial Mixing Zone It has been shown by Herwig (2) that overshoot at all convective boundaries, including the base of the convective envelope and the bottom of the He flash convection zone, strongly increases the efficiency of the third dredge-up at low core masses. This is required observationally in order to reproduce the observed C star luminosity function in the Magellanic Clouds (Richer 1981; Frogel et al. 199). Synthetic models of the AGB phase in which the third dredge-up parameter is derived observationally have demonstrated that typically efficient third dredge-up must take place at core masses as low as.58 M (Marigo et al. 1996), or at even lower core masses around.54 M according to a more recent analyzes by Marigo et al. (1999). This condition is not met by most stellar models, including those of Mowlavi (1999), who considers hydrodynamic overshoot only at the bottom of the convective envelope (see his Fig. 1b for a comparison of third dredge-up efficiencies found by different authors). The model grid of Herwig, Blöcker, & Driebe (2), which includes overshoot at all convective boundaries, includes cases (e.g., the 2 M, Z ¼ :1 case) that cover the low C star luminosity tail, as required by observations. The third dredge-up properties of AGB models are important for the s-process because third dredge-up is needed to bring the processed material to the surface. Hydrodynamic overshoot creates a partial mixing zone at the core-envelope interface with a continuous decrease of the H/ 12 C ratio from the envelope into the intershell layers (Herwig et al. 1997; Herwig 2). The global properties of the s-process can be reproduced with a partial mixing zone resulting from hydrodynamic overshoot (Goriely & Mowlavi 2; Lugaro & Herwig 21). The main features of the s-process overabundance distribution are mainly determined by the regions of the pocket that have the largest neutron exposures, and not so much by the detailed shape of the H/ 12 C profile within the partial mixing zone. Even if the treatment of hydrodynamic overshoot according to an exponentially decaying velocity field is not correct and the actual functional dependence of overshoot efficiency with depth is somewhat different, the s-process will most likely be affected only slightly as long as the H/ 12 C profile is somehow continuous. For example, Denissenkov & Tout (23) investigate gravity waves below the convective envelope as a cause for extra mixing to produce partial mixing of protons and 12 C. This mixing process is another way of looking at the mixing resulting from the perturbation of the convective boundary due to turbulence and leads to neutron exposures in the region close to that of previous models featuring a continuous decrease of the proton abundances into the 12 C-rich core. An additional effect is introduced in models that consider overshoot at all convective boundaries (Herwig 2). In these models the 12 C abundance in the intershell is about twice as large as that in models without overshoot at the base of the He-shell flash convection zone. Lugaro et al. (23b) have shown that the neutron exposure in the s-process layer is proportional to the 12 C abundance in the intershell. Hence, in models that consider overshoot at the base of the He-shell flash convection zone, the neutron

9 164 HERWIG, LANGER, & LUGARO Vol. 593 exposure in the s-process layer is higher than in models that do not include this overshoot How Much Overshoot? The initial computations of AGB stars with hydrodynamic overshoot were carried out with an exponential overshoot parameter of f ¼ :16, which was motivated by the efficiency derived from convective core overshoot of mainsequence stars. The effective mass of the partial mixing zone where the neutrons are efficiently released is confined within the region where the proton abundance follows 2 < log X p < 3 (for an intershell 12 C mass fraction of 2%; Goriely & Mowlavi 2). According to this criterion the mass of the s-process layer computed with f ¼ :16 is only 1 6 M, which is much smaller than required (see x 2). However, one overshoot efficiency parameter will not apply to all convective boundaries during all evolutionary phases. After Shaviv & Salpeter (1973) first considered the possibility of the convective overshoot, several studies have used a very simple prescription in which convective mixing was treated instantaneously and overshoot was simply a matter of extending the instantaneously mixed region by some fraction of the pressure scale height. In this approximation main-sequence core overshoot should extend by about.2h p (see, e.g., Schaller et al and references therein). Alongi et al. (1991) argued that overshoot of.7h p below the envelope of red giant stars could align the location of luminosity bump with observations. The two-dimensional radiation hydrodynamic simulations by Freytag et al. (1996) have shown that the shallow surface convection zone of white dwarfs has exponential overshoot mixing according to an overshoot parameter of f ¼ 1:, while the convection zone simulation of A stars shows f ¼ :25. For the oxygenburning layer in presupernova models, Asida & Arnett (2) found perturbations of the stable layers reaching 1H p beyond the formal convective boundary. Thus, there is ample indication that the overshoot efficiency is not the same at different convective boundaries. However, convective overshoot is not a stochastic process as long as the convective turnover timescale is shorter than the thermal timescale of the region that hosts the convective boundary. For similar convective boundaries one should expect a similar overshoot efficiency. This expectation is supported by two-dimensional hydrodynamical computations by Deupree (2), who showed that the core overshoot distance of ZAMS stars varies only mildly with stellar mass. Here we choose an exponential overshoot parameter for the hydrodynamic overshoot at the bottom of convective envelope of f ¼ :16. This larger overshoot is only applied during the third dredge-up phase. This has no major effect on the properties of the models, other than stretching the partial mixing zone and, consequently, the 13 Cand 14 N profiles in that layer over a larger mass range. The peak neutron exposure and the s-process abundance distribution in the partial mixing layer are not much changed. As a side effect the third dredge-up efficiency is slightly increased, by 2%. As mentioned in x 1, observations as well as the analysis of presolar meteoritic SiC grains suggest that stars with otherwise identical initial conditions have a range of s-process efficiencies. Such a range cannot be expected to result from overshoot since such a mechanism is not expected to be a stochastic process Neutron Production for the s-process in the Overshoot Model We model the abundance evolution in the partial mixing zone during the seventh interpulse phase of the 3 M, Z ¼ :2 sequence of Herwig et al. (2), with an overshoot efficiency f ¼ :16 during the third dredge-up phase. We use the postprocessing code (SBM6) described in x 3 because the computation of the stellar evolution does not include all the species and reactions needed to study the s-process. As initial conditions we use the thermodynamic and abundance profiles from the stellar evolution model at the end of the third dredge-up phase after the TP. These profiles are mapped to the equidistant, Lagrangian postprocessing grid and then evolved according to the stellar structure models at a series of times throughout the interpulse phase. We start the simulation with the partially mixed H/ 12 C zone of 1 4 M that has formed at the end of the third dredge-up phase as a result of time- and depth-dependent hydrodynamic overshoot (top panel, Fig. 6). In this model no mixing takes place during the interpulse phase. In the middle panel of Figure 6 the 13 C neutron source has started releasing neutrons, and up to 1% of the 13 C abundance has been consumed. In the upper part of the partial mixing zone, where 14 N dominates, the majority of neutrons are absorbed by the 14 N(n, p) 14 C reaction. This can be seen from the profile of 14 C. The maximum neutron density is 5:7 1 6 cm 3 mass fraction log(x) log(x), log(n n )-8 neutron exposure τ (mbarn -1 ) H 12 C C 14 N Nn 14 C m r /M sun Fig. 6. Abundance profiles in the partial mixing zone at three times during the seventh interpulse phase of the model with hydrodynamic overshoot and no rotation. Top: First postprocessing model after the end of the third dredge-up phase. Middle: 1% of 13 C is burned by 13 C(, n) 16 O. Bottom: Neutron exposure at end of the s-process in the partial mixing zone when the 13 C neutron source is exhausted.